ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Singularly Perturbed Cauchy Problem in the Presence of the Rational "simple" Pivot Point of the Limit Operator

### Author(s):

Alexandr G. Eliseev

Higher mathematics department, docent
National Research University "Moscow Power Engineering Institute",
Address: Krasnokazarmennaya 14, Moscow, 111250 Russia
Docent, PhD in physics and mathematics

eliseevag@mpei.ru

Tatiana A. Ratnikova

Higher mathematics department, docent
National Research University "Moscow Power Engineering Institute",
Address: Krasnokazarmennaya 14, Moscow, 111250 Russia

ratnikovata@mpei.ru

### Abstract:

Basing on the S. A. Lomov regularization method an asymptotic solution for a singularly perturbed Cauchy problem for the case when the stability conditions for the spectrum of the limit operator are violated is constructed. In particular, the problem with a “simple” pivot point when one eigenvalue at the initial moment of time has zero of arbitrary fractional order (the limit operator is discretely irreversible) is considered. This work is the development of the ideas described in the works of S.A. Lomov and A.G. Eliseev. Fractional pivot point in the simplest particular case was studied by the boundary function method by K.G. Kozhobekov and D.A. Tursunov. These problems have not been previously considered from the point of view of the regularization method. In the present paper on the basis of the theory of normal and unique solvability of iterative tasks elaborated by the authors, an algorithm for the regularization method is designed and substantiated, and an asymptotic solution of any order with respect to a small parameter is constructed.

### Keywords

• asymptotic solution
• boundary layer
• pivot point
• regularization method
• singularly perturbed Cauchy problem

### References:

1. Lomov, S. A. Vvedenie v obshyj teorij singuliajrnikh vozmyshenii [Introduction to the General Theory of Singular Perturbations]. Moscow, Nauka Publ., 1981. 400 p. (in Russian)
2. Eliseev, A. G., Lomov, S. A. [Theory of singular perturbations in the case of spectral singularities of the limit operator]. Matematicheskii sbornik, 1986; vol. 131, № 173: 544-557. (in Russian)
3. Lioville J. [Second memoir on the development of series functions of which various terms are subject to the same equation]. J. Math. Pure Appl. , 1837; vol. 2: 16-35. (in French)
4. Tursunov, D. A., Kozhobekov, K. G. [The asymptotics of solutions of singularly perturbed differential equations with fractional turning point]. Izvestiya Irkutskogo Gosudarstvennogo Universiteta, 2017; vol. 21: 108-121. (in Russian)